Suppose you were cast away on a desert island. You’re only allowed to take a maximum of eight known theorems with you, along with rudimentary results such as ZFC axioms together with ‘boring stuff’ such as mathematical induction over the naturals, commutativity of real multiplication and the least upper-bound property of . Everything else you have to prove yourself.
So, which eight theorems would you take with you on a desert island? It would be a waste of time to take the Baire category theorem, for instance; despite being extremely useful, it’s pretty trivial to prove. The same applies to the Dirichlet pigeonhole principle. On the other end of the spectrum, whilst Fermat’s Last Theorem is very difficult to prove, the end result is not nearly as useful as the machinery developed by Andrew Wiles in the course of solving this problem.
At the end of the post, there’s a ‘comments’ section for you to mention which theorems you would choose if marooned on a desert island. Here are my choices:
8. Density Hales-Jewett theorem
The ordinary Hales-Jewett theorem is quite straightforward to prove, using plenty of induction and the Dirichlet pigeonhole principle. On the other hand, the density analogue (c.f. Szemeredi’s theorem) is much deeper, requiring the entire Polymath project to establish a purely combinatorial proof (the original proof involved ergodic theory).
I’m not sure whether I’ve ever required the full force of Density Hales-Jewett, but Szemeredi’s theorem and ordinary Hales-Jewett have proved invaluable to me.
7. The abc theorem
Disclaimer: I don’t think that a second person is capable of understanding Mochizuki theory*, so this might still be classed as a conjecture until someone can peer-review the paper.
* Formerly known as inter-universal Teichmüller theory, although Doron Zeilberger makes a good argument that it should be renamed.
The abc conjecture is concerned with triples of coprime naturals, a < b < c, such that a + b = c. It states that the radical d (larger squarefree divisor) of abc cannot be much smaller than c. Specifically, for any ε > 0, we can only find finitely many examples where . This asserts that Beal’s conjecture holds in all but a finite number of cases, as does Fermat’s last theorem.
6. Max-flow min-cut theorem
I would have ranked this higher, except for the fact that it has a short elementary proof. The statement of the theorem is about networks, which are directed (multi-)graphs where each edge has a maximum capacity. A flow in a network is an assignment of nonnegative real values to each of the edges, such that they do not exceed the capacity, and that Kirchoff’s current law is obeyed at all vertices (with the exception of the source and sink vertices). The statement of the theorem is that the maximum flow attainable is equal to the minimum capacity of a cut (partitioning of the graph into two sets of vertices, so as to separate the source from the sink).
It can be used to prove Menger’s theorem in graph theory, Hall’s marriage theorem, Dilworth’s theorem on partially-ordered sets, and the Erdős-Szekeres theorem (although this also follows from Dilworth’s easier twin, Mirsky’s theorem).
5. Borsuk-Ulam theorem
This is informally stated as ‘two antipodal points on the Earth’s surface have the same temperature and pressure’. More generally, if we have a continuous function f from the n-sphere to , then we can find antipodal points x and y such that f(x) = f(y).
A corollary is the Brouwer fixed-point theorem, and all that that implies.
4. Gödel’s completeness theorem
This is one of the most beautiful and powerful theorems in mathematical logic. If a statement in first-order logic is consistent (i.e. we cannot prove a contradiction), then this asserts that there exists a finite or countable model. Consequently, we establish logical compactness: a set of first-order sentences has a model if and only if every finite subset has a model.
With the compactness theorem, the problem of computing the chromatic number of the plane is reduced to the more tractable problem of determining the maximum chromatic number of any finite unit-distance graph.
3. Every set can be well-ordered
This is equivalent to the axiom of choice, but much friendlier than the boring statement of AC involving a choice function. The main advantage is that it endows one with the power of transfinite induction over arbitrary sets, which is one of my favourite tools for proving theorems:
- Constructing an infinite game where neither player has a winning strategy;
- 2-colouring the plane such that there is no continuous monochromatic path;
- Proving that the direct product of two infinite sets has cardinality equal to the larger of the two sets (by Cantor normal form);
- Establishing Zorn’s lemma to show that any vector space has a basis…
2. Classification of finite simple groups
This is an impossible endeavour for a single individual to attempt, with the current proof being a 5000-page behemoth comprising many different papers. The Classification obviously reduces deep results such as the Feit-Thompson theorem (that there are no finite simple groups with odd composite order) and Burnside’s theorem to easy corollaries. (Of course, these theorems were almost certainly used in establishing the Classification, so this would be logically circular.)
It’s also a very beautiful theorem, with a vast wealth of exciting groups such as PSL(2,7), exceptional Lie groups over finite fields, and the Monster group. The Classification was also involved in proving theorems in areas outside group theory, such as establishing which graphs are 4- and 5-ultrahomogeneous.
1. Bezout’s theorem
This is a result in geometry, which says that if a degree-m and degree-n algebraic curve intersect in finitely many points, then they do so in at most mn points. Moreover, equality holds when the curves are on the complex projective plane, and we count intersections with the appropriate multiplicity.
An immediate corollary of this is the fundamental theorem of calculus, by taking one of the curves to be the line y = 0 and the other to be y = f(x) (where terms have been multiplied by appropriate powers of z to homogeneise them, and f is an arbitrary polynomial).
Also, it can be used to establish the Cayley-Bacharach theorem of cubic curves, which itself can prove the associativity of the elliptic curve group operation, Pascal’s theorem, Pappus’ theorem, and thus Desargues’ theorem.